Geometric numerical integration Robert McLachlan Professor of Applied Mathematics, Massey University VIC 2004 – p.1/26
What is geometric integration? A numerical method for a differential equation which inherits some property of the equation exactly. VIC 2004 – p.2/26
What is geometric integration? A numerical method for a differential equation which inherits some property of the equation exactly. The property should be possible to impose exactly, while still constraining the solution in some useful way. VIC 2004 – p.2/26
What is geometric integration? A numerical method for a differential equation which inherits some property of the equation exactly. The property should be possible to impose exactly, while still constraining the solution in some useful way. Examples: preserving first integrals, symmetries, phase space volume, symplecticity for Hamiltonian systems, reversibility, Lyapunov functions VIC 2004 – p.2/26
A basic example Earth (after a whole day) Earth (after half a day) Earth (initial position) Sun The leapfrog method Loup Verlet, 1967 VIC 2004 – p.3/26
Feynman’s Lectures on Physics VIC 2004 – p.4/26
Newton’s Principia , Book I, Theorem I VIC 2004 – p.5/26
The leapfrog method... is simple is fast is relatively accurate is time-reversible preserves momentum and angular momentum has no drift of energy preserves quasiperiodic orbits [KAM tori] produces qualitatively correct chaotic orbits is symplectic VIC 2004 – p.6/26
Channel & Scovel, 1990 VIC 2004 – p.7/26
What is symplecticity? Phase space T ∗ Q carries a symplectic 2-form ω Flow of Hamilton’s equations i X ω = − dH satisfies exp( tX ) ∗ ω = ω The leapfrog method also satisfies ϕ ∗ ω = ω VIC 2004 – p.8/26
What is symplecticity? B velocity impulse C A coast position VIC 2004 – p.9/26
Missed opportunities Freeman Dyson “I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.” VIC 2004 – p.10/26
The ice ages explained (Sverker Edvarddson 2002) VIC 2004 – p.11/26
Simulation of cold water 1 VIC 2004 – p.12/26
Simulation of cold water 2 VIC 2004 – p.12/26
Simulation of cold water 3 VIC 2004 – p.12/26
Simulation of cold water 4 VIC 2004 – p.12/26
Simulation of cold water 5 VIC 2004 – p.12/26
Simulation of cold water 6 VIC 2004 – p.12/26
Simulation of cold water 7 VIC 2004 – p.12/26
Simulation of cold water 8 VIC 2004 – p.12/26
Simulation of cold water 9 VIC 2004 – p.12/26
Simulation of cold water 10 VIC 2004 – p.12/26
Simulation of cold water 11 VIC 2004 – p.12/26
Simulation of cold water 12 VIC 2004 – p.12/26
Simulation of cold water 13 VIC 2004 – p.12/26
Simulation of cold water 14 VIC 2004 – p.12/26
Simulation of cold water 15 VIC 2004 – p.12/26
Simulation of cold water 15 VIC 2004 – p.12/26
Simulation of cold water 15 VIC 2004 – p.12/26
Backward error analysis Let G be a set of diffeomorphisms with a tangent space G at the identity. Let ϕ = 1 + (∆ t ) X + o (∆ t ) ∈ G . Then If ϕ ∈ C r then there exists � X ∈ G such that ϕ = exp(∆ t � X ) + O ((∆ t ) r ) If ϕ is analytic then ϕ = exp(∆ t � X ) + O ( e − a/ ∆ t ) . Therefore properties generic to G are (almost) preserved. For example, let G = symplectic maps, G =Hamiltonian vector fields. Then symplectic integrators have bounded energy errors for exponentially long times O ( e a/ ∆ t ) . VIC 2004 – p.13/26
Splitting methods Let X = � n i =1 X i with X, X i ∈ G . Use the integrators ϕ 1 (∆ t ) = exp(∆ tX 1 ) ◦ . . . exp(∆ tX n ) ϕ 2 (∆ t ) = ϕ 1 (∆ t/ 2) ϕ − 1 1 ( − ∆ t/ 2) z = (2 − 2 1 / 3 ) − 1 ϕ 4 (∆ t ) = ϕ 2 ( z ∆ t ) ϕ 2 ((1 − 2 z )∆ t ) ϕ 2 ( z ∆ t ) , where � � ∆ tX + O ((∆ t ) j +1 ) ϕ j = exp . Requires knowing an explicit form (generating function) for all X ∈ G a way of constructing lots of integrable X i ∈ G . the Baker-Campbell-Hausdorff formula � � A + B + 1 2 [ A, B ] + 1 exp( A ) exp( B ) = exp 6 [ A, [ A, B ]] + . . . VIC 2004 – p.14/26
Analysis of splitting methods Quickly leads to studying L ( X 1 , X 2 , . . . ) , the free Lie algebra generated by the X i . Simplest and most common case: split the Hamiltonian H = T + V = 1 2 pM ( q ) p + V ( q ) and study L ( T, V ) . This Lie algebra is not free, because T is quadratic in p , leading to [ V, [ V, [ V, T ]]] ≡ 0 . It is L P ( T, V ) , the Lie algebra of classical mechanics. VIC 2004 – p.15/26
Polynomial gradings Definition. A Lie algebra L is of class P (‘polynomially graded’) if it is graded, i.e. L P = � n ≥ 0 L n , and its homogeneous subspaces L n satisfy [ L n , L m ] ⊆ L n + m − 1 if n > 0 or m > 0 ; and [ L 0 , L 0 ] = 0 We call the grading of L its grading by degree. L P ( T, V ) also has the standard grading by order (= number of Lie brackets + 1), which satisfies L P = � m> 0 L m and [ L n , L m ] ⊆ L n + m . VIC 2004 – p.16/26
The Lie algebra of classical mechanics Theorem. L P ( T, V ) = Z ⊕ L ( T, [ T, Z ]) where Z = { V, [ V, [ V, T ]] , . . . } is an infinite set of degree–0 (potential energy) functions. P ( T, V ) ∼ 1 dim L n nα n , where α = 1 . 82542377420108 . . . is the entropy of classical mechanics. n 2 n (Witt 1936).) (Recall that in the free case, dim L n ( A, B ) ∼ 1 VIC 2004 – p.17/26
GF for CM 1 2.3 0.8 2 0.7 1.75 0.6 1.55 1.35 0.4 1 1.15 1 0.2 0.7 0.4 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 � � e x ( t ) � � where x ( t ) = � ∞ n =0 dim L n P ( T, V ) t n . VIC 2004 – p.18/26
Classification of dynamical systems Sets of dynamical systems can form a semigroup (examples: systems with Lyapunov functions, systems which contract volume) symmetric space closed under ( ϕ, ψ ) �→ ϕψ − 1 ϕ (example: maps with a reversing symmetry ϕ − 1 = RϕR − 1 ) group (examples: all diffeomorphisms, symplectic maps, volume-preserving maps) with linearizations to a Lie wedge, Lie triple, and Lie algebra respectively. VIC 2004 – p.19/26
The big picture 1. to find all groups, semigroups, and symmetric spaces of diffeomorphisms 2. to find their normal forms 3. to find a way to detect the structure in a given system 4. to study their relationships under intersection 5. to study the dynamics of their perturbations 6. to determine the characteristic dynamics and invariants of each class 7. to develop good (i.e. simple, fast, stable) integrators for each class VIC 2004 – p.20/26
What was known in 1895 all differential equations all "primitive" equations all conformal volume preserving "nonprimitive" equations volume conformal symplectic preserving symplectic contact VIC 2004 – p.21/26
What is known in 2003 Lie was right! all differential equations all "primitive" equations all conformal volume preserving "nonprimitive" equations volume conformal symplectic preserving symplectic contact VIC 2004 – p.22/26
Nonprimitive groups of diffeomorphisms preserve some foliation of phase space (leaves map to leaves) may preserve some (e.g. symplectic) structure on the leaves may preserve some structure on the space of leaves may have some complicated interaction between the leaves and the leaf space 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 y y y −0.5 −0.5 −0.5 −1 −1 −1 −1.5 −1.5 −1.5 −2 −2 −2 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x x x VIC 2004 – p.23/26
Example Let G be a Lie group acting on the Poisson manifold ( P, { , } ) . Let H : P → be a G -invariant Hamiltonian. Its flow preserves 4 foliations: 1. the level sets of H 2. the level sets of the momentum map of G 3. the symplectic leaves of P 4. the orbits of G More generally, a diffeomorphism ϕ on a manifold M can preserve any set of foliations, which must contain ∅ , M , and be closed under intersections and joins. That is, the set of foliations forms a lattice (generalizes Kodaira & Spencer 1961). VIC 2004 – p.24/26
Lattices with ≤ 3 foliations TM TM not foliate D 2 D D 1 0 0 not not modular distributive VIC 2004 – p.25/26
The big picture 1. to find all groups, semigroups, and symmetric spaces of diffeomorphisms 2. to find their normal forms 3. to find a way to detect the structure in a given system 4. to study their relationships under intersection 5. to study the dynamics of their perturbations 6. to determine the characteristic dynamics and invariants of each class 7. to develop good (i.e. simple, fast, stable) integrators for each class Thank you for your attention! VIC 2004 – p.26/26
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